{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:CAU32FEO3VXVFJ6YC2QAP5DJNJ","short_pith_number":"pith:CAU32FEO","canonical_record":{"source":{"id":"2506.22341","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-27T15:51:22Z","cross_cats_sorted":["math.CA","math.DS","math.GN"],"title_canon_sha256":"3cd8f06265c342612020be42eddaa841831f894b7d2e7de9888f8064fcda875e","abstract_canon_sha256":"dfbdfd01716151d592197bedc3959503a9a25a53c1f73300cd01479e8975e0e8"},"schema_version":"1.0"},"canonical_sha256":"1029bd148edd6f52a7d816a007f4696a6fa44b085956bd6f5f868505c6107634","source":{"kind":"arxiv","id":"2506.22341","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.22341","created_at":"2026-07-05T11:28:25Z"},{"alias_kind":"arxiv_version","alias_value":"2506.22341v1","created_at":"2026-07-05T11:28:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.22341","created_at":"2026-07-05T11:28:25Z"},{"alias_kind":"pith_short_12","alias_value":"CAU32FEO3VXV","created_at":"2026-07-05T11:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"CAU32FEO3VXVFJ6Y","created_at":"2026-07-05T11:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"CAU32FEO","created_at":"2026-07-05T11:28:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:CAU32FEO3VXVFJ6YC2QAP5DJNJ","target":"record","payload":{"canonical_record":{"source":{"id":"2506.22341","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-27T15:51:22Z","cross_cats_sorted":["math.CA","math.DS","math.GN"],"title_canon_sha256":"3cd8f06265c342612020be42eddaa841831f894b7d2e7de9888f8064fcda875e","abstract_canon_sha256":"dfbdfd01716151d592197bedc3959503a9a25a53c1f73300cd01479e8975e0e8"},"schema_version":"1.0"},"canonical_sha256":"1029bd148edd6f52a7d816a007f4696a6fa44b085956bd6f5f868505c6107634","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:28:25.507109Z","signature_b64":"LGaaUZ9+XM6F2gAmRcLw1jxb+V/sfQ2mPCeWAXtB9SFMD8cMmhUGmbGY/yD4Ndp6+ZYvaViwgagjbhYByFTVBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1029bd148edd6f52a7d816a007f4696a6fa44b085956bd6f5f868505c6107634","last_reissued_at":"2026-07-05T11:28:25.506636Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:28:25.506636Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.22341","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:28:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RRDd3ZyV5oEAqXZjtYEyKGF8EMcPrPv6LxGqVllLMMpVor5wmnA+OmfpOF3ALcWNOjJGlt8dsYeC3zEq8t+XDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T08:15:41.534085Z"},"content_sha256":"dc0eb6e0854c99bbbff2be7a09de5d6e3658bb0a08a8d851936598ac9d489382","schema_version":"1.0","event_id":"sha256:dc0eb6e0854c99bbbff2be7a09de5d6e3658bb0a08a8d851936598ac9d489382"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:CAU32FEO3VXVFJ6YC2QAP5DJNJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the complexity of upper frequently hypercyclic vectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.DS","math.GN"],"primary_cat":"math.FA","authors_text":"Paolo Leonetti, Szymon Glab","submitted_at":"2025-06-27T15:51:22Z","abstract_excerpt":"Given a continuous linear operator $T:X\\to X$, where $X$ is a topological vector space, let $\\mathrm{UFHC}(T)$ be the set of upper frequently hypercyclic vectors, that is, the set of vectors $x \\in X$ such that $\\{n \\in \\omega: T^nx \\in U\\}$ has positive upper asymptotic density for all nonempty open sets $U\\subseteq X$. It is known that $\\mathrm{UFHC}(T)$ is a $G_{\\delta\\sigma\\delta}$-set which is either empty or contains a dense $G_{\\delta}$-set. Using a purely topological proof, we improve it by showing that $\\mathrm{UFHC}(T)$ is always a $G_{\\delta\\sigma}$-set.\n  Bonilla and Grosse-Erdmann"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.22341","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.22341/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:28:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YyuofbTQaSEDaYMQJBjWmWfrtPOpdHTBz1CvDHyXfEp4VutVH9acCLCC5VH/DfhdLiejJ6gDKxfi/zpCIR2hAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T08:15:41.534608Z"},"content_sha256":"ac3a57c694c4607d79c5c95e16269cfdec28223cfe534f14556296b6266f92f2","schema_version":"1.0","event_id":"sha256:ac3a57c694c4607d79c5c95e16269cfdec28223cfe534f14556296b6266f92f2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CAU32FEO3VXVFJ6YC2QAP5DJNJ/bundle.json","state_url":"https://pith.science/pith/CAU32FEO3VXVFJ6YC2QAP5DJNJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CAU32FEO3VXVFJ6YC2QAP5DJNJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-09T08:15:41Z","links":{"resolver":"https://pith.science/pith/CAU32FEO3VXVFJ6YC2QAP5DJNJ","bundle":"https://pith.science/pith/CAU32FEO3VXVFJ6YC2QAP5DJNJ/bundle.json","state":"https://pith.science/pith/CAU32FEO3VXVFJ6YC2QAP5DJNJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CAU32FEO3VXVFJ6YC2QAP5DJNJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:CAU32FEO3VXVFJ6YC2QAP5DJNJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"dfbdfd01716151d592197bedc3959503a9a25a53c1f73300cd01479e8975e0e8","cross_cats_sorted":["math.CA","math.DS","math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-27T15:51:22Z","title_canon_sha256":"3cd8f06265c342612020be42eddaa841831f894b7d2e7de9888f8064fcda875e"},"schema_version":"1.0","source":{"id":"2506.22341","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.22341","created_at":"2026-07-05T11:28:25Z"},{"alias_kind":"arxiv_version","alias_value":"2506.22341v1","created_at":"2026-07-05T11:28:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.22341","created_at":"2026-07-05T11:28:25Z"},{"alias_kind":"pith_short_12","alias_value":"CAU32FEO3VXV","created_at":"2026-07-05T11:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"CAU32FEO3VXVFJ6Y","created_at":"2026-07-05T11:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"CAU32FEO","created_at":"2026-07-05T11:28:25Z"}],"graph_snapshots":[{"event_id":"sha256:ac3a57c694c4607d79c5c95e16269cfdec28223cfe534f14556296b6266f92f2","target":"graph","created_at":"2026-07-05T11:28:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2506.22341/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given a continuous linear operator $T:X\\to X$, where $X$ is a topological vector space, let $\\mathrm{UFHC}(T)$ be the set of upper frequently hypercyclic vectors, that is, the set of vectors $x \\in X$ such that $\\{n \\in \\omega: T^nx \\in U\\}$ has positive upper asymptotic density for all nonempty open sets $U\\subseteq X$. It is known that $\\mathrm{UFHC}(T)$ is a $G_{\\delta\\sigma\\delta}$-set which is either empty or contains a dense $G_{\\delta}$-set. Using a purely topological proof, we improve it by showing that $\\mathrm{UFHC}(T)$ is always a $G_{\\delta\\sigma}$-set.\n  Bonilla and Grosse-Erdmann","authors_text":"Paolo Leonetti, Szymon Glab","cross_cats":["math.CA","math.DS","math.GN"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-27T15:51:22Z","title":"On the complexity of upper frequently hypercyclic vectors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.22341","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:dc0eb6e0854c99bbbff2be7a09de5d6e3658bb0a08a8d851936598ac9d489382","target":"record","created_at":"2026-07-05T11:28:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"dfbdfd01716151d592197bedc3959503a9a25a53c1f73300cd01479e8975e0e8","cross_cats_sorted":["math.CA","math.DS","math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-27T15:51:22Z","title_canon_sha256":"3cd8f06265c342612020be42eddaa841831f894b7d2e7de9888f8064fcda875e"},"schema_version":"1.0","source":{"id":"2506.22341","kind":"arxiv","version":1}},"canonical_sha256":"1029bd148edd6f52a7d816a007f4696a6fa44b085956bd6f5f868505c6107634","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1029bd148edd6f52a7d816a007f4696a6fa44b085956bd6f5f868505c6107634","first_computed_at":"2026-07-05T11:28:25.506636Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:28:25.506636Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LGaaUZ9+XM6F2gAmRcLw1jxb+V/sfQ2mPCeWAXtB9SFMD8cMmhUGmbGY/yD4Ndp6+ZYvaViwgagjbhYByFTVBw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:28:25.507109Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.22341","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc0eb6e0854c99bbbff2be7a09de5d6e3658bb0a08a8d851936598ac9d489382","sha256:ac3a57c694c4607d79c5c95e16269cfdec28223cfe534f14556296b6266f92f2"],"state_sha256":"e7eb893634e12ba4b22b2135703445c56397e166279c4ee57d0b60ab5ac2de71"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/wjx/XIe8ZIWGObz0EjSBQoV/asbv+6ufz43cVWw3da0kxNrX9lz2JPrDc5bH56pgiow0Oykv9Ta5Jq/6vllAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T08:15:41.543664Z","bundle_sha256":"f36a781f208403cc8a30da9f402615f29b3ee5fd25ffcc60f294a24a7b192cb1"}}